What would be done with decommissioned naval vessels during the Age of Sail?

A common life cycle for a warship was to begin its career as either a purpose built warship or converted merchantman. Once it was older and either obsolete or no longer fit for war it might become a storeship for supplies or a training vessel. Sometime after that it might be moved to a harbor and permanently moored for use as a supply shed or barracks. Some might even be converted into temporary buildings. Finally when the hull was completely used up she would be scrapped and the useful salvage reused, metal fittings and fasteners being among the more desirable salvage.

The first USS New Hampshire, built in 1819, never saw combat as a warship, but is a good example of how a hull too old for war might see continued use. She was built, stored until the Civil War, then used as a depot ship and training vessel for the next 40 years.

Alabama, one of “nine ships to rate not less than 74 guns each” authorized by Congress 29 April 1816, was laid down by the Portsmouth Navy Yard, N.H., in June 1819, the year the State of Alabama was admitted to the Union. Though ready for launch by 1825, she remained on the stocks for preservation until needed during the Civil War. Renamed New Hampshire 28 October 1863, she was launched 23 April 1864, fitted out as a stores and depot ship of the South Atlantic Blockading Squadron; and commissioned 13 May 1864, Commodore Henry K. Thatcher in command.

New Hampshire sailed from Portsmouth 15 June and relieved sister ship Vermont 29 July 1864 as store and depot ship at Port Royal, S.C., and served there through the end of the Civil War. She returned to Norfolk 8 June 1866, serving as a receiving ship there until 10 May 1876 when she sailed back to Port Royal. She resumed duty at Norfolk in 1881 but soon shifted to Newport, R.I. She became flagship of Commodore Stephen B. Luce’s newly formed Apprentice Training Squadron, marking the commencement of an effective apprentice training program for the Navy.

New Hampshire was towed from Newport to New London, Conn., in 1891 and was receiving ship there until decommissioned 5 June 1892. The following year she was loaned as a training ship for the New York State Naval Militia which was to furnish nearly a thousand officers and men to the Navy during the Spanish-American War.

New Hampshire was renamed Granite State 30 November 1904 to free the name New Hampshire for a newly authorized battleship. Stationed in the Hudson River, she continued training service throughout the years leading to World War I when State naval militia were practically the only trained and equipped men available to the Navy for immediate service. They were mustered into the Navy as National Naval Volunteers. Secretary of the Navy Josephus Daniels wrote in his Our Navy at War: “Never again will men dare riducule the Volunteer, the Reservist, the man who in a national crisis lays aside civilian duty to become a soldier or sailor—They fought well. They died well. They have left in deeds and words a record that will be an inspiration to unborn generations.”

Granite State served the New York State Militia until she caught fire and sank at her pier in the Hudson River 23 May 1921. Her hull was sold for salvage 19 August 1921 to the Mulholland Machinery Corp. Refloated in July 1922, she and was taken in tow to the Bay of Fundy. The towline parted during a storm, she again caught fire and sank off Half Way Rock in Massachusetts Bay.

This is a great overview. I'd also add that in the 16th-18th century English navy, obsolete ships were sometimes completely dismantled and used to rebuild entirely new ships in what shipwrights called a "rebuild" or "great rebuild." This was essentially a bureaucratic dodge used to get around the fact that Parliament sometimes went years or decades without voting money for new ship construction (in a previous post, I mentioned that Parliament didn't vote new shipbuilding money for the entire period 1696-1745).

I'd also mention that in the British service at least, obsolete ships could also be transformed into prison hulks, to store prisoners of war (most famously from the Napoleonic period, but they were used earlier and later than that). Conditions aboard the prison hulks could be described as horrific as best.

Another common use for a hulk (a ship that's no longer capable of going to sea) would be as a sheer hulk - a floating crane to be used in refitting and repairing other ships. In particular it would be useful in extracting old and setting new masts.

This was essentially a bureaucratic dodge used to get around the fact that Parliament sometimes went years or decades without voting money for new ship construction

Something similar goes on nowadays (at least in California) where permits to tear down a house and build a new one are hard to get - but ones for "renovations" are fairly simple. This leads to "renovations" where they remove all but like a half-dozen studs (usually at the corners of the old house) and then build from there.

That can, for instance, lead to a set of three small California Bungalows being "renovated" into a two-story apartment building...

Privateer ships were custom built. So were all other wooden ships, so they were not alone in this fact. Many, perhaps most, would be converted merchant ships. The would-be privateers would select a swift looking merchant ship and outfit it with light cannon and vastly oversized crews if putting together a dedicated privateer. The oversize crew was for boarding and putting a prize crew aboard victims. Other privateers were simply normal merchant captains who placed a few guns on their ships and acquired a letter of marque in case they happened across a ship belonging to a rival power in wartime. Really well off privateers might have access to a small warship such as Drake’s Golden Hind.

I suspect the gist of your question though is whether any type of ship was especially common among privateers. There is one type of ship that did become closely associated with the privateer lifestyle. These ships were known as Baltimore Clippers. These are not clipper ships. Baltimore clippers are much smaller and occur earlier. They were medium sized vessels, generally with two masts, and a sharp wineglass hull. This gave them great speed and agility which was ideal for privateering. They were very popular among American Privateers during the war of 1812.

The sources online are somewhat poor, so you would be best to find a copy of The Baltimore Clipper by Howard I. Chapelle (I haven’t read this one, but Chapelle knew his stuff and was a major figure in the Smithsonian boat collection so I trust that it is solid).

Some private vessels would also be sold to the military to be used for training purposes. For example, the Thermopalylae was a clipper ship (1868) used for the tea trade and later used for the wool trade. She was sold off to Portugal to be used as a naval training ship. When she reached the end of her life (1907) the Portugese navy towed her out and topedoed her. Interesting side note, the Thermopylae actually raced and beat the famous Cutty Sark and still holds the fastest trip under sail.